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Harmonic Analysis and Hartley Transform Associated with the Generalized Differential-Difference Operator

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Abstract

We consider a new differential-difference operator \(\Lambda \) on the real line. We study the harmonic analysis associated with this operator and we prove the Bochner and Bochner-Schwartz theorems, the Plancherel formula and the inversion theorems for the Hartley transform associated to the operator \(\Lambda \).

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References

  1. Bracewell, R.N.: The Hartley Transform. Oxford University Press, New York (1986)

    MATH  Google Scholar 

  2. Hartley, R.V.L.: A more symmetrical Fourier analysis applied to transmission problems. Proc. IRE. 30, 144–150 (1942)

    Article  MathSciNet  MATH  Google Scholar 

  3. Tuan, V-K., Yakubovich, S-B.: A criterion for the unitarity of a two-sided integral transformation (in Russian). Ukr. Mat. Zh. 44(5), 697–699 (translation in Ukr. Math. J. 44(5), 630–632 (1992))

  4. Hai, N.-T., Yakubovich, S., Wimp, J.: Multidimensional Watson transforms. Int. J. Math. Stat. Sci. 1(1), 105–119 (1992)

    MathSciNet  MATH  Google Scholar 

  5. Yakubovich, S.: On the half-Hartley transform, its iteration and compositions with Fourier transforms. J. Integral Equ. Appl. 26(4), 581–608 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Yakubovich, S.: The Plancherel, Titchmarsh and convolution theorems for the half-Hartley transform. Integral Transforms Spec. Funct. 25(10), 836–848 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Trimèche, K.: Inversion of the J.L. Lions transmutation operators using generalized wavelets. Appl. Comput. Harmon. Anal. 4, 97–112 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  8. Trimèche, K.: The transmutation operators relating to a Dunkl Type operator on \({\mathbb{R}}\) and their positivity. Mediterr. J. Math. 12(2), 349–369 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bloom, W.R., Xu, Z.: Fourier transforms of Schwartz functions on Chébli-Trimèche hypergroups. Mon. Math. 125, 89–109 (1998)

    Article  MATH  Google Scholar 

  10. Schwartz, L.: Théorie des Distributions. Hermann, Paris (1966)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, College of Sciences, Taibah University, PO BOX 30002, Medina, Saudi Arabia

    Hatem Mejjaoli

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  1. Hatem Mejjaoli
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Correspondence to Hatem Mejjaoli.

Additional information

Communicated by Matthias Langer.

I dedicate this paper to the Emeritus Professor Khalifa Trimèche on his 70 Birthday.

The author is deeply indebted to the referees for providing constructive comments and helps in improving the contents of this article.

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Mejjaoli, H. Harmonic Analysis and Hartley Transform Associated with the Generalized Differential-Difference Operator. Complex Anal. Oper. Theory 11, 1351–1369 (2017). https://doi.org/10.1007/s11785-016-0585-9

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  • DOI: https://doi.org/10.1007/s11785-016-0585-9

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